A particle is displaced from the point `A(5, -5, -7)` to the point `B(6, 2, -2)` under the action of forces `P_(1) =10i-j+11k, P_(2) = 4i+5j+6k, P_(3) = -2i+j-9k,` then the work done is

Options

A. 81
B. 85
C. 87
D. none

Answer

To solve the problem of finding the work done when a particle is displaced from point A to point B under the action of three forces, we can follow these steps:

### Step 1: Determine the Displacement Vector
The displacement vector \( \mathbf{D} \) from point A to point B can be calculated using the formula:
\[ \mathbf{D} = \mathbf{B} - \mathbf{A} \]
Given points:
- \( A(5, -5, -7) \)
- \( B(6, 2, -2) \)

Calculating the components:
\[ \mathbf{D} = (6 - 5)\mathbf{i} + (2 - (-5))\mathbf{j} + (-2 - (-7))\mathbf{k} \]
\[ \mathbf{D} = 1\mathbf{i} + (2 + 5)\mathbf{j} + (7)\mathbf{k} \]
\[ \mathbf{D} = 1\mathbf{i} + 7\mathbf{j} + 5\mathbf{k} \]

### Step 2: Calculate the Resultant Force Vector
The resultant force vector \( \mathbf{F} \) is the sum of the three given force vectors:
\[ \mathbf{P_1} = 10\mathbf{i} - \mathbf{j} + 11\mathbf{k} \]
\[ \mathbf{P_2} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k} \]
\[ \mathbf{P_3} = -2\mathbf{i} + \mathbf{j} - 9\mathbf{k} \]

Calculating the resultant:
\[ \mathbf{F} = \mathbf{P_1} + \mathbf{P_2} + \mathbf{P_3} \]
\[ \mathbf{F} = (10 + 4 - 2)\mathbf{i} + (-1 + 5 + 1)\mathbf{j} + (11 + 6 - 9)\mathbf{k} \]
\[ \mathbf{F} = 12\mathbf{i} + 5\mathbf{j} + 8\mathbf{k} \]

### Step 3: Calculate the Work Done
The work done \( W \) is given by the dot product of the force vector \( \mathbf{F} \) and the displacement vector \( \mathbf{D} \):
\[ W = \mathbf{F} \cdot \mathbf{D} \]
Calculating the dot product:
\[ W = (12\mathbf{i} + 5\mathbf{j} + 8\mathbf{k}) \cdot (1\mathbf{i} + 7\mathbf{j} + 5\mathbf{k}) \]
\[ W = (12 \cdot 1) + (5 \cdot 7) + (8 \cdot 5) \]
\[ W = 12 + 35 + 40 \]
\[ W = 87 \]

### Final Answer
The work done by the forces is \( \boxed{87} \).

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